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<a href="_hypervolume_subset_selection2_d_8h.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a id="l00001" name="l00001"></a><span class="lineno">    1</span><span class="comment">/*!</span></div>
<div class="line"><a id="l00002" name="l00002"></a><span class="lineno">    2</span><span class="comment"> *</span></div>
<div class="line"><a id="l00003" name="l00003"></a><span class="lineno">    3</span><span class="comment"> * \author      O.Krause</span></div>
<div class="line"><a id="l00004" name="l00004"></a><span class="lineno">    4</span><span class="comment"> * \date        2016</span></div>
<div class="line"><a id="l00005" name="l00005"></a><span class="lineno">    5</span><span class="comment"> *</span></div>
<div class="line"><a id="l00006" name="l00006"></a><span class="lineno">    6</span><span class="comment"> *</span></div>
<div class="line"><a id="l00007" name="l00007"></a><span class="lineno">    7</span><span class="comment"> * \par Copyright 1995-2017 Shark Development Team</span></div>
<div class="line"><a id="l00008" name="l00008"></a><span class="lineno">    8</span><span class="comment"> * </span></div>
<div class="line"><a id="l00009" name="l00009"></a><span class="lineno">    9</span><span class="comment"> * &lt;BR&gt;&lt;HR&gt;</span></div>
<div class="line"><a id="l00010" name="l00010"></a><span class="lineno">   10</span><span class="comment"> * This file is part of Shark.</span></div>
<div class="line"><a id="l00011" name="l00011"></a><span class="lineno">   11</span><span class="comment"> * &lt;https://shark-ml.github.io/Shark/&gt;</span></div>
<div class="line"><a id="l00012" name="l00012"></a><span class="lineno">   12</span><span class="comment"> * </span></div>
<div class="line"><a id="l00013" name="l00013"></a><span class="lineno">   13</span><span class="comment"> * Shark is free software: you can redistribute it and/or modify</span></div>
<div class="line"><a id="l00014" name="l00014"></a><span class="lineno">   14</span><span class="comment"> * it under the terms of the GNU Lesser General Public License as published </span></div>
<div class="line"><a id="l00015" name="l00015"></a><span class="lineno">   15</span><span class="comment"> * by the Free Software Foundation, either version 3 of the License, or</span></div>
<div class="line"><a id="l00016" name="l00016"></a><span class="lineno">   16</span><span class="comment"> * (at your option) any later version.</span></div>
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<div class="line"><a id="l00021" name="l00021"></a><span class="lineno">   21</span><span class="comment"> * GNU Lesser General Public License for more details.</span></div>
<div class="line"><a id="l00022" name="l00022"></a><span class="lineno">   22</span><span class="comment"> * </span></div>
<div class="line"><a id="l00023" name="l00023"></a><span class="lineno">   23</span><span class="comment"> * You should have received a copy of the GNU Lesser General Public License</span></div>
<div class="line"><a id="l00024" name="l00024"></a><span class="lineno">   24</span><span class="comment"> * along with Shark.  If not, see &lt;http://www.gnu.org/licenses/&gt;.</span></div>
<div class="line"><a id="l00025" name="l00025"></a><span class="lineno">   25</span><span class="comment"> *</span></div>
<div class="line"><a id="l00026" name="l00026"></a><span class="lineno">   26</span><span class="comment"> */</span></div>
<div class="line"><a id="l00027" name="l00027"></a><span class="lineno">   27</span><span class="preprocessor">#ifndef SHARK_ALGORITHMS_DIRECTSEARCH_HYPERVOLUMESUBSETSELECTION_2D_H</span></div>
<div class="line"><a id="l00028" name="l00028"></a><span class="lineno">   28</span><span class="preprocessor">#define SHARK_ALGORITHMS_DIRECTSEARCH_HYPERVOLUMESUBSETSELECTION_2D_H</span></div>
<div class="line"><a id="l00029" name="l00029"></a><span class="lineno">   29</span> </div>
<div class="line"><a id="l00030" name="l00030"></a><span class="lineno">   30</span><span class="preprocessor">#include &lt;<a class="code" href="_base_8h.html">shark/LinAlg/Base.h</a>&gt;</span></div>
<div class="line"><a id="l00031" name="l00031"></a><span class="lineno">   31</span> </div>
<div class="line"><a id="l00032" name="l00032"></a><span class="lineno">   32</span><span class="preprocessor">#include &lt;algorithm&gt;</span></div>
<div class="line"><a id="l00033" name="l00033"></a><span class="lineno">   33</span><span class="preprocessor">#include &lt;vector&gt;</span></div>
<div class="line"><a id="l00034" name="l00034"></a><span class="lineno">   34</span><span class="preprocessor">#include &lt;deque&gt;</span></div>
<div class="line"><a id="l00035" name="l00035"></a><span class="lineno">   35</span> </div>
<div class="line"><a id="l00036" name="l00036"></a><span class="lineno">   36</span><span class="keyword">namespace </span><a class="code hl_namespace" href="namespaceshark.html" title="AbstractMultiObjectiveOptimizer.">shark</a> {<span class="comment"></span></div>
<div class="line"><a id="l00037" name="l00037"></a><span class="lineno">   37</span><span class="comment">/// \brief Implementation of the exact hypervolume subset selection algorithm in 2 dimensions.</span></div>
<div class="line"><a id="l00038" name="l00038"></a><span class="lineno">   38</span><span class="comment">///</span></div>
<div class="line"><a id="l00039" name="l00039"></a><span class="lineno">   39</span><span class="comment">/// This algorithm solves the problem of selecting a subset of points with largest hypervolume in 2D.</span></div>
<div class="line"><a id="l00040" name="l00040"></a><span class="lineno">   40</span><span class="comment">/// The algorithm has complexity n (k+log(n)) </span></div>
<div class="line"><a id="l00041" name="l00041"></a><span class="lineno">   41</span><span class="comment">///</span></div>
<div class="line"><a id="l00042" name="l00042"></a><span class="lineno">   42</span><span class="comment">/// While this algorithm accepts fronts with dominated points in it, the caller has to ensure</span></div>
<div class="line"><a id="l00043" name="l00043"></a><span class="lineno">   43</span><span class="comment">/// that after domination checks there are at least as many points left as there are to select. The</span></div>
<div class="line"><a id="l00044" name="l00044"></a><span class="lineno">   44</span><span class="comment">/// Algorithm will throw an exception otherwise.</span></div>
<div class="line"><a id="l00045" name="l00045"></a><span class="lineno">   45</span><span class="comment">///</span></div>
<div class="line"><a id="l00046" name="l00046"></a><span class="lineno">   46</span><span class="comment">/// This can easily be ensured by removing the nondominated points prior to calling this function.</span></div>
<div class="line"><a id="l00047" name="l00047"></a><span class="lineno">   47</span><span class="comment">///</span></div>
<div class="line"><a id="l00048" name="l00048"></a><span class="lineno">   48</span><span class="comment">/// The algorithm is described in:</span></div>
<div class="line"><a id="l00049" name="l00049"></a><span class="lineno">   49</span><span class="comment">/// Bringmann, Karl, Tobias Friedrich, and Patrick Klitzke. </span></div>
<div class="line"><a id="l00050" name="l00050"></a><span class="lineno">   50</span><span class="comment">/// &quot;Two-dimensional subset selection for hypervolume and epsilon-indicator.&quot;</span></div>
<div class="line"><a id="l00051" name="l00051"></a><span class="lineno">   51</span><span class="comment">/// Proceedings of the 2014 conference on Genetic and evolutionary computation. </span></div>
<div class="line"><a id="l00052" name="l00052"></a><span class="lineno">   52</span><span class="comment">/// ACM, 2014.  </span></div>
<div class="line"><a id="l00053" name="l00053"></a><span class="lineno">   53</span><span class="comment">/// (although it is not very helpful)</span></div>
<div class="foldopen" id="foldopen00054" data-start="{" data-end="};">
<div class="line"><a id="l00054" name="l00054"></a><span class="lineno"><a class="line" href="structshark_1_1_hypervolume_subset_selection2_d.html">   54</a></span><span class="comment"></span><span class="keyword">struct </span><a class="code hl_struct" href="structshark_1_1_hypervolume_subset_selection2_d.html" title="Implementation of the exact hypervolume subset selection algorithm in 2 dimensions.">HypervolumeSubsetSelection2D</a> {</div>
<div class="line"><a id="l00055" name="l00055"></a><span class="lineno">   55</span><span class="keyword">private</span>:</div>
<div class="line"><a id="l00056" name="l00056"></a><span class="lineno">   56</span>    </div>
<div class="line"><a id="l00057" name="l00057"></a><span class="lineno">   57</span>    <span class="keyword">struct </span>Point{</div>
<div class="line"><a id="l00058" name="l00058"></a><span class="lineno">   58</span>        Point(){}</div>
<div class="line"><a id="l00059" name="l00059"></a><span class="lineno">   59</span> </div>
<div class="line"><a id="l00060" name="l00060"></a><span class="lineno">   60</span>        Point(<span class="keywordtype">double</span> f1, <span class="keywordtype">double</span> f2, std::size_t index)</div>
<div class="line"><a id="l00061" name="l00061"></a><span class="lineno">   61</span>        : f1(f1)</div>
<div class="line"><a id="l00062" name="l00062"></a><span class="lineno">   62</span>        , f2(f2)</div>
<div class="line"><a id="l00063" name="l00063"></a><span class="lineno">   63</span>        , index(index)</div>
<div class="line"><a id="l00064" name="l00064"></a><span class="lineno">   64</span>        , selected(<span class="keyword">false</span>)</div>
<div class="line"><a id="l00065" name="l00065"></a><span class="lineno">   65</span>        {}</div>
<div class="line"><a id="l00066" name="l00066"></a><span class="lineno">   66</span>            </div>
<div class="line"><a id="l00067" name="l00067"></a><span class="lineno">   67</span>        <span class="keywordtype">bool</span> operator&lt;(Point <span class="keyword">const</span>&amp; rhs)<span class="keyword"> const</span>{<span class="comment">//for lexicographic sorting</span></div>
<div class="line"><a id="l00068" name="l00068"></a><span class="lineno">   68</span>            <span class="keywordflow">if</span> (f1 &lt; rhs.f1) <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
<div class="line"><a id="l00069" name="l00069"></a><span class="lineno">   69</span>            <span class="keywordflow">if</span> (f1 &gt; rhs.f1) <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
<div class="line"><a id="l00070" name="l00070"></a><span class="lineno">   70</span>            <span class="keywordflow">return</span> (f2 &lt; rhs.f1);</div>
<div class="line"><a id="l00071" name="l00071"></a><span class="lineno">   71</span>        }</div>
<div class="line"><a id="l00072" name="l00072"></a><span class="lineno">   72</span>        </div>
<div class="line"><a id="l00073" name="l00073"></a><span class="lineno">   73</span>        <span class="keywordtype">double</span> f1;</div>
<div class="line"><a id="l00074" name="l00074"></a><span class="lineno">   74</span>        <span class="keywordtype">double</span> f2;</div>
<div class="line"><a id="l00075" name="l00075"></a><span class="lineno">   75</span>        std::size_t index;</div>
<div class="line"><a id="l00076" name="l00076"></a><span class="lineno">   76</span>        <span class="keywordtype">bool</span> selected;</div>
<div class="line"><a id="l00077" name="l00077"></a><span class="lineno">   77</span>    };</div>
<div class="line"><a id="l00078" name="l00078"></a><span class="lineno">   78</span>    <span class="comment"></span></div>
<div class="line"><a id="l00079" name="l00079"></a><span class="lineno">   79</span><span class="comment">    ///\brief Linear function a*x+b where a is stored in first and b is stored in section.</span></div>
<div class="line"><a id="l00080" name="l00080"></a><span class="lineno">   80</span><span class="comment">    ///</span></div>
<div class="line"><a id="l00081" name="l00081"></a><span class="lineno">   81</span><span class="comment">    /// The linear function also stores an index to uniquely identify it.</span></div>
<div class="line"><a id="l00082" name="l00082"></a><span class="lineno">   82</span><span class="comment">    ///</span></div>
<div class="line"><a id="l00083" name="l00083"></a><span class="lineno">   83</span><span class="comment">    /// Linear functions are used in the algorithm to represent the </span></div>
<div class="line"><a id="l00084" name="l00084"></a><span class="lineno">   84</span><span class="comment">    /// volume of a given set of points under the change of reference point.</span></div>
<div class="line"><a id="l00085" name="l00085"></a><span class="lineno">   85</span><span class="comment">    /// more formally, let H^l_i be the volume of a set of points of size l with largest</span></div>
<div class="line"><a id="l00086" name="l00086"></a><span class="lineno">   86</span><span class="comment">    /// x-value at the point (x_i,y_i) and reference point x_i(thus H^l_i can only use points</span></div>
<div class="line"><a id="l00087" name="l00087"></a><span class="lineno">   87</span><span class="comment">    /// 1,...,i). </span></div>
<div class="line"><a id="l00088" name="l00088"></a><span class="lineno">   88</span><span class="comment">    /// Then for x&gt;x_i we have</span></div>
<div class="line"><a id="l00089" name="l00089"></a><span class="lineno">   89</span><span class="comment">    /// f_i^l(x) = H_i^l+ y_i(x_i-x)=-x*y_i+y_i*x_i+H = a*x+b. </span></div>
<div class="line"><a id="l00090" name="l00090"></a><span class="lineno">   90</span><span class="comment">    /// Later the algorithm will use an upper envelope over a set of those functions</span></div>
<div class="line"><a id="l00091" name="l00091"></a><span class="lineno">   91</span><span class="comment">    /// to decide which points to add to the sets until the size of the sets is k.</span></div>
<div class="line"><a id="l00092" name="l00092"></a><span class="lineno">   92</span><span class="comment">    ///</span></div>
<div class="line"><a id="l00093" name="l00093"></a><span class="lineno">   93</span><span class="comment">    /// for this application the stored index is the same as index i of the point stated above.</span></div>
<div class="line"><a id="l00094" name="l00094"></a><span class="lineno">   94</span><span class="comment"></span>    <span class="keyword">struct </span>LinearFunction{</div>
<div class="line"><a id="l00095" name="l00095"></a><span class="lineno">   95</span>    </div>
<div class="line"><a id="l00096" name="l00096"></a><span class="lineno">   96</span>        <span class="keywordtype">double</span> a;</div>
<div class="line"><a id="l00097" name="l00097"></a><span class="lineno">   97</span>        <span class="keywordtype">double</span> b;</div>
<div class="line"><a id="l00098" name="l00098"></a><span class="lineno">   98</span>        std::size_t index;</div>
<div class="line"><a id="l00099" name="l00099"></a><span class="lineno">   99</span>        </div>
<div class="line"><a id="l00100" name="l00100"></a><span class="lineno">  100</span>        LinearFunction(<span class="keywordtype">double</span> a, <span class="keywordtype">double</span> b, std::size_t index = 0):a(a), b(b), index(index){}</div>
<div class="line"><a id="l00101" name="l00101"></a><span class="lineno">  101</span>        LinearFunction(){}</div>
<div class="line"><a id="l00102" name="l00102"></a><span class="lineno">  102</span>        </div>
<div class="line"><a id="l00103" name="l00103"></a><span class="lineno">  103</span>        <span class="keywordtype">double</span> eval(<span class="keywordtype">double</span> x)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00104" name="l00104"></a><span class="lineno">  104</span>            <span class="keywordflow">return</span> a*x + b;</div>
<div class="line"><a id="l00105" name="l00105"></a><span class="lineno">  105</span>        }</div>
<div class="line"><a id="l00106" name="l00106"></a><span class="lineno">  106</span>    };</div>
<div class="line"><a id="l00107" name="l00107"></a><span class="lineno">  107</span><span class="comment"></span> </div>
<div class="line"><a id="l00108" name="l00108"></a><span class="lineno">  108</span><span class="comment">    /// \brief Returns the intersection of two linear functions</span></div>
<div class="line"><a id="l00109" name="l00109"></a><span class="lineno">  109</span><span class="comment"></span>    <span class="keywordtype">double</span> Intersection(LinearFunction f1, LinearFunction f2)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00110" name="l00110"></a><span class="lineno">  110</span>        <span class="keywordflow">return</span> (f2.b - f1.b) / (f1.a - f2.a);</div>
<div class="line"><a id="l00111" name="l00111"></a><span class="lineno">  111</span>    }</div>
<div class="line"><a id="l00112" name="l00112"></a><span class="lineno">  112</span>    </div>
<div class="line"><a id="l00113" name="l00113"></a><span class="lineno">  113</span>    <span class="comment"></span></div>
<div class="line"><a id="l00114" name="l00114"></a><span class="lineno">  114</span><span class="comment">    /// \brief  Calculates for each given x the maximum among the functions f, i.e. the upper envelope of f.</span></div>
<div class="line"><a id="l00115" name="l00115"></a><span class="lineno">  115</span><span class="comment">    /// </span></div>
<div class="line"><a id="l00116" name="l00116"></a><span class="lineno">  116</span><span class="comment">    /// Algorithm 2 in the paper. Complexity O(n)</span></div>
<div class="line"><a id="l00117" name="l00117"></a><span class="lineno">  117</span><span class="comment">    /// given a set of functions f_1...f_n, ordered by slope such that f_1.a &lt; f_2.a&lt;...&lt;f_n.a and points with x-coordinate x_1&lt;...&lt;x_n</span></div>
<div class="line"><a id="l00118" name="l00118"></a><span class="lineno">  118</span><span class="comment">    /// computes h_i = max_{1 &lt;= j &lt;= i} f_j(x_i) for i=1,...,n as well as the index of the function leading to the value h_i</span></div>
<div class="line"><a id="l00119" name="l00119"></a><span class="lineno">  119</span><span class="comment"></span>    std::pair&lt;std::vector&lt;double&gt;,std::vector&lt;std::size_t&gt; &gt; upperEnvelope(</div>
<div class="line"><a id="l00120" name="l00120"></a><span class="lineno">  120</span>        std::vector&lt;LinearFunction&gt;<span class="keyword">const</span>&amp; functions,</div>
<div class="line"><a id="l00121" name="l00121"></a><span class="lineno">  121</span>        std::vector&lt;Point&gt; <span class="keyword">const</span>&amp; points</div>
<div class="line"><a id="l00122" name="l00122"></a><span class="lineno">  122</span>    )<span class="keyword">const</span>{</div>
<div class="line"><a id="l00123" name="l00123"></a><span class="lineno">  123</span>        <a class="code hl_define" href="_exception_8h.html#a73abb5049a0168d72a48e72dda41708b">SHARK_ASSERT</a>(functions.size() == points.size());</div>
<div class="line"><a id="l00124" name="l00124"></a><span class="lineno">  124</span>        std::size_t n = points.size();</div>
<div class="line"><a id="l00125" name="l00125"></a><span class="lineno">  125</span>        std::vector&lt;double&gt; h(n);</div>
<div class="line"><a id="l00126" name="l00126"></a><span class="lineno">  126</span>        std::vector&lt;std::size_t&gt; chosen(n);</div>
<div class="line"><a id="l00127" name="l00127"></a><span class="lineno">  127</span>        std::deque&lt;LinearFunction&gt; s;</div>
<div class="line"><a id="l00128" name="l00128"></a><span class="lineno">  128</span> </div>
<div class="line"><a id="l00129" name="l00129"></a><span class="lineno">  129</span>        <span class="comment">// This is the original algorithm 2 as in the paper. Even if the paper looks at maximum</span></div>
<div class="line"><a id="l00130" name="l00130"></a><span class="lineno">  130</span>        <span class="comment">// hypervolume where domination is given when one point has LARGER function</span></div>
<div class="line"><a id="l00131" name="l00131"></a><span class="lineno">  131</span>        <span class="comment">// values as the other, In section 3.2 they transform the problem to a problem</span></div>
<div class="line"><a id="l00132" name="l00132"></a><span class="lineno">  132</span>        <span class="comment">// where domination is given by SMALLER function values and then accordingly</span></div>
<div class="line"><a id="l00133" name="l00133"></a><span class="lineno">  133</span>        <span class="comment">// give the algorithm for this type. They just give the transformation but do not say</span></div>
<div class="line"><a id="l00134" name="l00134"></a><span class="lineno">  134</span>        <span class="comment">// what the transformation does so it is not clear until you implement it.</span></div>
<div class="line"><a id="l00135" name="l00135"></a><span class="lineno">  135</span>        <span class="comment">//</span></div>
<div class="line"><a id="l00136" name="l00136"></a><span class="lineno">  136</span>        <span class="comment">// This is a super confusing part of the paper, please kids, do not be like Bringmann et al.</span></div>
<div class="line"><a id="l00137" name="l00137"></a><span class="lineno">  137</span>        <span class="comment">// Keep it simple, stupid. Sometimes an additional index does not hurt.</span></div>
<div class="line"><a id="l00138" name="l00138"></a><span class="lineno">  138</span>        <span class="comment">//</span></div>
<div class="line"><a id="l00139" name="l00139"></a><span class="lineno">  139</span>        <span class="comment">//the algorithm works by inserting functions f_1 to f_i one-by-one, figuring out which functions</span></div>
<div class="line"><a id="l00140" name="l00140"></a><span class="lineno">  140</span>        <span class="comment">// are dominated (not being part of the upper envelope) and removing all functions which for</span></div>
<div class="line"><a id="l00141" name="l00141"></a><span class="lineno">  141</span>        <span class="comment">// function values x_i,x_i+1,... are already smaller than one of the other function.</span></div>
<div class="line"><a id="l00142" name="l00142"></a><span class="lineno">  142</span>        <span class="comment">// using the ordering relations given the set s contains the function ordered by (current) function value.</span></div>
<div class="line"><a id="l00143" name="l00143"></a><span class="lineno">  143</span>        <span class="comment">// so after iteration i we can just extract the largest function value for x_i by looking at the first element of s.</span></div>
<div class="line"><a id="l00144" name="l00144"></a><span class="lineno">  144</span>        <span class="keywordflow">for</span> (std::size_t i = 0; i != n; ++i) {</div>
<div class="line"><a id="l00145" name="l00145"></a><span class="lineno">  145</span> </div>
<div class="line"><a id="l00146" name="l00146"></a><span class="lineno">  146</span>            <span class="comment">// remove dominated functions.</span></div>
<div class="line"><a id="l00147" name="l00147"></a><span class="lineno">  147</span>            <span class="comment">// as we push back into s,</span></div>
<div class="line"><a id="l00148" name="l00148"></a><span class="lineno">  148</span>            <span class="comment">// at the end of s are the functions with largest slope.</span></div>
<div class="line"><a id="l00149" name="l00149"></a><span class="lineno">  149</span>            <span class="comment">// therefore if we have the last two elements as s_-1 and s_-2 and the new</span></div>
<div class="line"><a id="l00150" name="l00150"></a><span class="lineno">  150</span>            <span class="comment">// function f, knowing that the intersection of s_-1 and f is smaller than the intersection</span></div>
<div class="line"><a id="l00151" name="l00151"></a><span class="lineno">  151</span>            <span class="comment">// of s_-1 and s_-2 means that s_-1 is dominated  by s_-2 and f and thus can be removed.</span></div>
<div class="line"><a id="l00152" name="l00152"></a><span class="lineno">  152</span>            <span class="keywordflow">while</span> (s.size() &gt; 1 ) {</div>
<div class="line"><a id="l00153" name="l00153"></a><span class="lineno">  153</span>                </div>
<div class="line"><a id="l00154" name="l00154"></a><span class="lineno">  154</span>                <span class="keywordtype">double</span> d1 = Intersection(functions[i], s.end()[-1]);</div>
<div class="line"><a id="l00155" name="l00155"></a><span class="lineno">  155</span>                <span class="keywordtype">double</span> d2 = Intersection(s.end()[-2], s.end()[-1]);</div>
<div class="line"><a id="l00156" name="l00156"></a><span class="lineno">  156</span> </div>
<div class="line"><a id="l00157" name="l00157"></a><span class="lineno">  157</span>                <span class="keywordflow">if</span> (d1 &lt;= d2 || std::abs(d1-d2) &lt; 1.e-10) {<span class="comment">//check for numeric stability</span></div>
<div class="line"><a id="l00158" name="l00158"></a><span class="lineno">  158</span>                    s.pop_back();</div>
<div class="line"><a id="l00159" name="l00159"></a><span class="lineno">  159</span>                } <span class="keywordflow">else</span> {</div>
<div class="line"><a id="l00160" name="l00160"></a><span class="lineno">  160</span>                    <span class="keywordflow">break</span>;</div>
<div class="line"><a id="l00161" name="l00161"></a><span class="lineno">  161</span>                }</div>
<div class="line"><a id="l00162" name="l00162"></a><span class="lineno">  162</span>            }</div>
<div class="line"><a id="l00163" name="l00163"></a><span class="lineno">  163</span>            <span class="comment">//include the new function and store its index.</span></div>
<div class="line"><a id="l00164" name="l00164"></a><span class="lineno">  164</span>            s.push_back(functions[i]);</div>
<div class="line"><a id="l00165" name="l00165"></a><span class="lineno">  165</span>            s.back().index = i;</div>
<div class="line"><a id="l00166" name="l00166"></a><span class="lineno">  166</span>            <span class="comment">// at the beginning of s are the functions with smallest slope</span></div>
<div class="line"><a id="l00167" name="l00167"></a><span class="lineno">  167</span>            <span class="comment">// if the first function in s has a smaller function value for the current </span></div>
<div class="line"><a id="l00168" name="l00168"></a><span class="lineno">  168</span>            <span class="comment">// x_i than the second function,</span></div>
<div class="line"><a id="l00169" name="l00169"></a><span class="lineno">  169</span>            <span class="comment">// we can safely remove it as it can not be part of the envelope any more</span></div>
<div class="line"><a id="l00170" name="l00170"></a><span class="lineno">  170</span>            <span class="comment">// (We are only looking at function values &gt;=x from now on and thus the larger slope domintes)</span></div>
<div class="line"><a id="l00171" name="l00171"></a><span class="lineno">  171</span>            <span class="keywordflow">while</span> (s.size() &gt; 1) {</div>
<div class="line"><a id="l00172" name="l00172"></a><span class="lineno">  172</span>                <span class="keywordtype">double</span> d1 = s[0].eval(points[i].f1);</div>
<div class="line"><a id="l00173" name="l00173"></a><span class="lineno">  173</span>                <span class="keywordtype">double</span> d2 = s[1].eval(points[i].f1);</div>
<div class="line"><a id="l00174" name="l00174"></a><span class="lineno">  174</span> </div>
<div class="line"><a id="l00175" name="l00175"></a><span class="lineno">  175</span>                <span class="keywordflow">if</span> (d1 &lt; d2 || std::abs(d1-d2) &lt; 1.e-10) {</div>
<div class="line"><a id="l00176" name="l00176"></a><span class="lineno">  176</span>                    s.pop_front();</div>
<div class="line"><a id="l00177" name="l00177"></a><span class="lineno">  177</span>                } <span class="keywordflow">else</span> {</div>
<div class="line"><a id="l00178" name="l00178"></a><span class="lineno">  178</span>                    <span class="keywordflow">break</span>;</div>
<div class="line"><a id="l00179" name="l00179"></a><span class="lineno">  179</span>                }</div>
<div class="line"><a id="l00180" name="l00180"></a><span class="lineno">  180</span>            }</div>
<div class="line"><a id="l00181" name="l00181"></a><span class="lineno">  181</span>            <span class="comment">//assign maximum</span></div>
<div class="line"><a id="l00182" name="l00182"></a><span class="lineno">  182</span>            <span class="comment">//the functions in s are ordered by function value  </span></div>
<div class="line"><a id="l00183" name="l00183"></a><span class="lineno">  183</span>            <span class="comment">// the function with the largest function value is currently at the front</span></div>
<div class="line"><a id="l00184" name="l00184"></a><span class="lineno">  184</span>            h[i] = s[0].eval(points[i].f1);</div>
<div class="line"><a id="l00185" name="l00185"></a><span class="lineno">  185</span>            chosen[i] = s[0].index;</div>
<div class="line"><a id="l00186" name="l00186"></a><span class="lineno">  186</span>        }</div>
<div class="line"><a id="l00187" name="l00187"></a><span class="lineno">  187</span>        <span class="keywordflow">return</span> std::make_pair(std::move(h),std::move(chosen));</div>
<div class="line"><a id="l00188" name="l00188"></a><span class="lineno">  188</span>    }</div>
<div class="line"><a id="l00189" name="l00189"></a><span class="lineno">  189</span>    </div>
<div class="line"><a id="l00190" name="l00190"></a><span class="lineno">  190</span>    <span class="comment"></span></div>
<div class="line"><a id="l00191" name="l00191"></a><span class="lineno">  191</span><span class="comment">    /// Fast calculation O(n*k) for the hypervolume selection problem. </span></div>
<div class="line"><a id="l00192" name="l00192"></a><span class="lineno">  192</span><span class="comment">    /// for the selected points, it sets selected=true.</span></div>
<div class="line"><a id="l00193" name="l00193"></a><span class="lineno">  193</span><span class="comment"></span>    <span class="keywordtype">void</span> hypSSP(std::vector&lt;Point&gt;&amp; front,std::size_t k)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00194" name="l00194"></a><span class="lineno">  194</span>        <a class="code hl_define" href="_exception_8h.html#adce1f80097c69010f5eab2618fa2e971">SHARK_RUNTIME_CHECK</a>( k &gt; 0, <span class="stringliteral">&quot;k must be non-zero&quot;</span>);</div>
<div class="line"><a id="l00195" name="l00195"></a><span class="lineno">  195</span>        <a class="code hl_define" href="_exception_8h.html#adce1f80097c69010f5eab2618fa2e971">SHARK_RUNTIME_CHECK</a>( k &lt;= front.size(), <span class="stringliteral">&quot;The front must have at least k nondominated points&quot;</span>);</div>
<div class="line"><a id="l00196" name="l00196"></a><span class="lineno">  196</span>        </div>
<div class="line"><a id="l00197" name="l00197"></a><span class="lineno">  197</span>        std::size_t n = front.size();</div>
<div class="line"><a id="l00198" name="l00198"></a><span class="lineno">  198</span>        std::vector&lt;LinearFunction&gt; functions(n);</div>
<div class="line"><a id="l00199" name="l00199"></a><span class="lineno">  199</span>        </div>
<div class="line"><a id="l00200" name="l00200"></a><span class="lineno">  200</span>        std::vector&lt;std::vector&lt;std::size_t&gt; &gt; chosen;</div>
<div class="line"><a id="l00201" name="l00201"></a><span class="lineno">  201</span>        std::vector&lt;double&gt;  h(n,0.0);</div>
<div class="line"><a id="l00202" name="l00202"></a><span class="lineno">  202</span>        <span class="keywordflow">for</span>(std::size_t j=0; j != k-1; ++j) {<span class="comment">//compute until k-1 elements are chosen</span></div>
<div class="line"><a id="l00203" name="l00203"></a><span class="lineno">  203</span>            <span class="keywordflow">for</span>(std::size_t i=0; i != n; ++i ) {</div>
<div class="line"><a id="l00204" name="l00204"></a><span class="lineno">  204</span>                functions[i] = LinearFunction( -front[i].f2, front[i].f1* front[i].f2 + h[i]);</div>
<div class="line"><a id="l00205" name="l00205"></a><span class="lineno">  205</span>            }</div>
<div class="line"><a id="l00206" name="l00206"></a><span class="lineno">  206</span>            <span class="keyword">auto</span> result = upperEnvelope(functions, front);</div>
<div class="line"><a id="l00207" name="l00207"></a><span class="lineno">  207</span>            h = result.first;</div>
<div class="line"><a id="l00208" name="l00208"></a><span class="lineno">  208</span>            chosen.push_back(result.second);</div>
<div class="line"><a id="l00209" name="l00209"></a><span class="lineno">  209</span>        }</div>
<div class="line"><a id="l00210" name="l00210"></a><span class="lineno">  210</span>        </div>
<div class="line"><a id="l00211" name="l00211"></a><span class="lineno">  211</span>        <span class="comment">//choose the last element by simply iterating over all elements</span></div>
<div class="line"><a id="l00212" name="l00212"></a><span class="lineno">  212</span>        std::size_t currentIndex = 0;</div>
<div class="line"><a id="l00213" name="l00213"></a><span class="lineno">  213</span>        <span class="keywordtype">double</span> res = -1;</div>
<div class="line"><a id="l00214" name="l00214"></a><span class="lineno">  214</span>        <span class="keywordflow">for</span>(std::size_t i=0; i != n; ++i ) {</div>
<div class="line"><a id="l00215" name="l00215"></a><span class="lineno">  215</span>            LinearFunction f(-front[i].f2, front[i].f1*front[i].f2 + h[i]);</div>
<div class="line"><a id="l00216" name="l00216"></a><span class="lineno">  216</span>            <span class="keywordflow">if</span>(f.eval(0)  &gt; res) {</div>
<div class="line"><a id="l00217" name="l00217"></a><span class="lineno">  217</span>                res = f.eval(0);</div>
<div class="line"><a id="l00218" name="l00218"></a><span class="lineno">  218</span>                currentIndex = i;</div>
<div class="line"><a id="l00219" name="l00219"></a><span class="lineno">  219</span>            }</div>
<div class="line"><a id="l00220" name="l00220"></a><span class="lineno">  220</span>        }</div>
<div class="line"><a id="l00221" name="l00221"></a><span class="lineno">  221</span>        front[currentIndex].selected = <span class="keyword">true</span>;</div>
<div class="line"><a id="l00222" name="l00222"></a><span class="lineno">  222</span>        <span class="comment">//iterate backwards to reconstruct chosen indizes</span></div>
<div class="line"><a id="l00223" name="l00223"></a><span class="lineno">  223</span>        <span class="keywordflow">for</span>(<span class="keyword">auto</span> pos = chosen.rbegin(); pos != chosen.rend(); ++pos){</div>
<div class="line"><a id="l00224" name="l00224"></a><span class="lineno">  224</span>            currentIndex = (*pos)[currentIndex];</div>
<div class="line"><a id="l00225" name="l00225"></a><span class="lineno">  225</span>            front[currentIndex].selected = <span class="keyword">true</span>;</div>
<div class="line"><a id="l00226" name="l00226"></a><span class="lineno">  226</span>        }</div>
<div class="line"><a id="l00227" name="l00227"></a><span class="lineno">  227</span>    }</div>
<div class="line"><a id="l00228" name="l00228"></a><span class="lineno">  228</span>    </div>
<div class="line"><a id="l00229" name="l00229"></a><span class="lineno">  229</span>    <span class="keyword">template</span>&lt;<span class="keyword">typename</span> Set&gt;</div>
<div class="line"><a id="l00230" name="l00230"></a><span class="lineno">  230</span>    std::vector&lt;Point&gt; createFront(Set <span class="keyword">const</span>&amp; points, <span class="keywordtype">double</span> refX, <span class="keywordtype">double</span> refY)<span class="keyword">const</span>{</div>
<div class="line"><a id="l00231" name="l00231"></a><span class="lineno">  231</span>        <span class="comment">//copy points using the new reference frame with refPoint at (0,0). also store original index for later</span></div>
<div class="line"><a id="l00232" name="l00232"></a><span class="lineno">  232</span>        std::vector&lt;Point&gt; front;</div>
<div class="line"><a id="l00233" name="l00233"></a><span class="lineno">  233</span>        <span class="keywordflow">for</span>(std::size_t i = 0; i != points.size(); ++i){</div>
<div class="line"><a id="l00234" name="l00234"></a><span class="lineno">  234</span>            front.emplace_back(points[i](0) - refX, points[i](1) - refY,i);</div>
<div class="line"><a id="l00235" name="l00235"></a><span class="lineno">  235</span>        }</div>
<div class="line"><a id="l00236" name="l00236"></a><span class="lineno">  236</span>        std::sort(front.begin(),front.end());<span class="comment">//sort lexicographically</span></div>
<div class="line"><a id="l00237" name="l00237"></a><span class="lineno">  237</span>        <span class="comment">//erase dominated points</span></div>
<div class="line"><a id="l00238" name="l00238"></a><span class="lineno">  238</span>        <span class="keyword">auto</span> newEnd = std::unique(front.begin(),front.end(),[](Point <span class="keyword">const</span>&amp; x, Point <span class="keyword">const</span>&amp; y){</div>
<div class="line"><a id="l00239" name="l00239"></a><span class="lineno">  239</span>            return y.f2 &gt;= x.f2;<span class="comment">//by lexikographic sort we already have y.f1 &gt;= x.f1</span></div>
<div class="line"><a id="l00240" name="l00240"></a><span class="lineno">  240</span>        });</div>
<div class="line"><a id="l00241" name="l00241"></a><span class="lineno">  241</span>        front.erase(newEnd,front.end());</div>
<div class="line"><a id="l00242" name="l00242"></a><span class="lineno">  242</span>        <span class="keywordflow">return</span> front;</div>
<div class="line"><a id="l00243" name="l00243"></a><span class="lineno">  243</span>    }</div>
<div class="line"><a id="l00244" name="l00244"></a><span class="lineno">  244</span><span class="keyword">public</span>:<span class="comment"></span></div>
<div class="line"><a id="l00245" name="l00245"></a><span class="lineno">  245</span><span class="comment">    /// \brief Executes the algorithm.</span></div>
<div class="line"><a id="l00246" name="l00246"></a><span class="lineno">  246</span><span class="comment">    /// While this algorithm in general accepts fronts with dominated points in it, the caller has to ensure</span></div>
<div class="line"><a id="l00247" name="l00247"></a><span class="lineno">  247</span><span class="comment">    /// that after domination checks there are at least as many points left as there are to select. The</span></div>
<div class="line"><a id="l00248" name="l00248"></a><span class="lineno">  248</span><span class="comment">    /// Algorithm will throw an exception otherwise.</span></div>
<div class="line"><a id="l00249" name="l00249"></a><span class="lineno">  249</span><span class="comment">    ///</span></div>
<div class="line"><a id="l00250" name="l00250"></a><span class="lineno">  250</span><span class="comment">    /// This can easily be ensured by removing the nondominated points prior to calling this function.</span></div>
<div class="line"><a id="l00251" name="l00251"></a><span class="lineno">  251</span><span class="comment">    /// \param [in] points The set \f$S\f$ of points to select</span></div>
<div class="line"><a id="l00252" name="l00252"></a><span class="lineno">  252</span><span class="comment">    /// \param [out] selected set of the same size as the set of points indicating whether the point is selected (1) or not (0)</span></div>
<div class="line"><a id="l00253" name="l00253"></a><span class="lineno">  253</span><span class="comment">    /// \param [in] k number of points to select. Must be lrger than 0</span></div>
<div class="line"><a id="l00254" name="l00254"></a><span class="lineno">  254</span><span class="comment">    /// \param [in] refPoint The reference point \f$\vec{r} \in \mathbb{R}^2\f$ for the hypervolume calculation, needs to fulfill: \f$ \forall s \in S: s \preceq \vec{r}\f$. .</span></div>
<div class="line"><a id="l00255" name="l00255"></a><span class="lineno">  255</span><span class="comment"></span>    <span class="keyword">template</span>&lt;<span class="keyword">typename</span> Set, <span class="keyword">typename</span> SelectedSet, <span class="keyword">typename</span> VectorType &gt;</div>
<div class="foldopen" id="foldopen00256" data-start="{" data-end="}">
<div class="line"><a id="l00256" name="l00256"></a><span class="lineno"><a class="line" href="structshark_1_1_hypervolume_subset_selection2_d.html#aa0aa8bb2bc003bc32d6a636895009ae4">  256</a></span>    <span class="keywordtype">void</span> <a class="code hl_function" href="structshark_1_1_hypervolume_subset_selection2_d.html#aa0aa8bb2bc003bc32d6a636895009ae4" title="Executes the algorithm. While this algorithm in general accepts fronts with dominated points in it,...">operator()</a>( Set <span class="keyword">const</span>&amp; points, SelectedSet&amp; selected, std::size_t k, <a class="code hl_typedef" href="_c_svm_linear_8cpp.html#ab106d665148183a2dc94dcf8716c9203">VectorType</a> <span class="keyword">const</span>&amp; refPoint){</div>
<div class="line"><a id="l00257" name="l00257"></a><span class="lineno">  257</span>        <a class="code hl_define" href="_exception_8h.html#a42a6a50e4d06c00d60fbca5333f40768">SIZE_CHECK</a>(points.size() == selected.size());</div>
<div class="line"><a id="l00258" name="l00258"></a><span class="lineno">  258</span>        <a class="code hl_define" href="_exception_8h.html#adce1f80097c69010f5eab2618fa2e971">SHARK_RUNTIME_CHECK</a>(k &gt; 0, <span class="stringliteral">&quot;k must be &gt;0&quot;</span>);</div>
<div class="line"><a id="l00259" name="l00259"></a><span class="lineno">  259</span>        <a class="code hl_define" href="_exception_8h.html#adce1f80097c69010f5eab2618fa2e971">SHARK_RUNTIME_CHECK</a>( k &lt;= points.size(), <span class="stringliteral">&quot;the number of points must be larger than k&quot;</span>);</div>
<div class="line"><a id="l00260" name="l00260"></a><span class="lineno">  260</span>        <a class="code hl_define" href="_exception_8h.html#a42a6a50e4d06c00d60fbca5333f40768">SIZE_CHECK</a>( points.begin()-&gt;size() == 2 );</div>
<div class="line"><a id="l00261" name="l00261"></a><span class="lineno">  261</span>        <a class="code hl_define" href="_exception_8h.html#a42a6a50e4d06c00d60fbca5333f40768">SIZE_CHECK</a>( refPoint.size() == 2 );</div>
<div class="line"><a id="l00262" name="l00262"></a><span class="lineno">  262</span>        </div>
<div class="line"><a id="l00263" name="l00263"></a><span class="lineno">  263</span>        <span class="keywordflow">for</span>(<span class="keyword">auto</span>&amp;&amp; s: selected)</div>
<div class="line"><a id="l00264" name="l00264"></a><span class="lineno">  264</span>            s = <span class="keyword">false</span>;</div>
<div class="line"><a id="l00265" name="l00265"></a><span class="lineno">  265</span>        </div>
<div class="line"><a id="l00266" name="l00266"></a><span class="lineno">  266</span>        std::vector&lt;Point&gt; front = createFront(points, refPoint(0), refPoint(1));</div>
<div class="line"><a id="l00267" name="l00267"></a><span class="lineno">  267</span>        </div>
<div class="line"><a id="l00268" name="l00268"></a><span class="lineno">  268</span>        <span class="comment">//find the optimal set in the front. afterwards selected points have selected=true</span></div>
<div class="line"><a id="l00269" name="l00269"></a><span class="lineno">  269</span>        hypSSP(front,k);</div>
<div class="line"><a id="l00270" name="l00270"></a><span class="lineno">  270</span>        <span class="comment">//mark selected points in the original front</span></div>
<div class="line"><a id="l00271" name="l00271"></a><span class="lineno">  271</span>        <span class="keywordflow">for</span>(Point <span class="keyword">const</span>&amp; point: front){</div>
<div class="line"><a id="l00272" name="l00272"></a><span class="lineno">  272</span>            <span class="keywordflow">if</span>(point.selected){</div>
<div class="line"><a id="l00273" name="l00273"></a><span class="lineno">  273</span>                selected[point.index] = <span class="keyword">true</span>;</div>
<div class="line"><a id="l00274" name="l00274"></a><span class="lineno">  274</span>            }</div>
<div class="line"><a id="l00275" name="l00275"></a><span class="lineno">  275</span>        }</div>
<div class="line"><a id="l00276" name="l00276"></a><span class="lineno">  276</span>    }</div>
</div>
<div class="line"><a id="l00277" name="l00277"></a><span class="lineno">  277</span>    <span class="comment"></span></div>
<div class="line"><a id="l00278" name="l00278"></a><span class="lineno">  278</span><span class="comment">    /// \brief Executes the algorithm.</span></div>
<div class="line"><a id="l00279" name="l00279"></a><span class="lineno">  279</span><span class="comment">    ///</span></div>
<div class="line"><a id="l00280" name="l00280"></a><span class="lineno">  280</span><span class="comment">    /// This version does not use a reference point. instead the extreme points are always kept which  implicitely defines a reference point</span></div>
<div class="line"><a id="l00281" name="l00281"></a><span class="lineno">  281</span><span class="comment">    /// that after domination checks there are at least as many points left as there are to select. The</span></div>
<div class="line"><a id="l00282" name="l00282"></a><span class="lineno">  282</span><span class="comment">    /// Algorithm will throw an exception otherwise.</span></div>
<div class="line"><a id="l00283" name="l00283"></a><span class="lineno">  283</span><span class="comment">    ///</span></div>
<div class="line"><a id="l00284" name="l00284"></a><span class="lineno">  284</span><span class="comment">    /// This can easily be ensured by removing the nondominated points prior to calling this function.</span></div>
<div class="line"><a id="l00285" name="l00285"></a><span class="lineno">  285</span><span class="comment">    ///</span></div>
<div class="line"><a id="l00286" name="l00286"></a><span class="lineno">  286</span><span class="comment">    /// \param [in] points The set \f$S\f$ of points to select</span></div>
<div class="line"><a id="l00287" name="l00287"></a><span class="lineno">  287</span><span class="comment">    /// \param [out] selected set of the same size as the set of points indicating whether the point is selected (1) or not (0)</span></div>
<div class="line"><a id="l00288" name="l00288"></a><span class="lineno">  288</span><span class="comment">    /// \param [in] k number of points to select, must be larger or equal 2</span></div>
<div class="line"><a id="l00289" name="l00289"></a><span class="lineno">  289</span><span class="comment"></span>    <span class="keyword">template</span>&lt;<span class="keyword">typename</span> Set, <span class="keyword">typename</span> SelectedSet&gt;</div>
<div class="foldopen" id="foldopen00290" data-start="{" data-end="}">
<div class="line"><a id="l00290" name="l00290"></a><span class="lineno"><a class="line" href="structshark_1_1_hypervolume_subset_selection2_d.html#a52fe369d85589e88caa8fde88615d253">  290</a></span>    <span class="keywordtype">void</span> <a class="code hl_function" href="structshark_1_1_hypervolume_subset_selection2_d.html#a52fe369d85589e88caa8fde88615d253" title="Executes the algorithm.">operator()</a>( Set <span class="keyword">const</span>&amp; points, SelectedSet&amp; selected, std::size_t k){</div>
<div class="line"><a id="l00291" name="l00291"></a><span class="lineno">  291</span>        <a class="code hl_define" href="_exception_8h.html#a42a6a50e4d06c00d60fbca5333f40768">SIZE_CHECK</a>(points.size() == selected.size());</div>
<div class="line"><a id="l00292" name="l00292"></a><span class="lineno">  292</span>        <a class="code hl_define" href="_exception_8h.html#adce1f80097c69010f5eab2618fa2e971">SHARK_RUNTIME_CHECK</a>( k &gt;= 2, <span class="stringliteral">&quot;k must be larger or equal 2&quot;</span>);</div>
<div class="line"><a id="l00293" name="l00293"></a><span class="lineno">  293</span>        <a class="code hl_define" href="_exception_8h.html#adce1f80097c69010f5eab2618fa2e971">SHARK_RUNTIME_CHECK</a>( k &lt;= points.size(), <span class="stringliteral">&quot;the number of points mjust be larger than k&quot;</span>);</div>
<div class="line"><a id="l00294" name="l00294"></a><span class="lineno">  294</span>        <a class="code hl_define" href="_exception_8h.html#a42a6a50e4d06c00d60fbca5333f40768">SIZE_CHECK</a>(points.size() == selected.size());</div>
<div class="line"><a id="l00295" name="l00295"></a><span class="lineno">  295</span>        <a class="code hl_define" href="_exception_8h.html#a42a6a50e4d06c00d60fbca5333f40768">SIZE_CHECK</a>( points.begin()-&gt;size() == 2 );</div>
<div class="line"><a id="l00296" name="l00296"></a><span class="lineno">  296</span>        </div>
<div class="line"><a id="l00297" name="l00297"></a><span class="lineno">  297</span>        <span class="keywordflow">for</span>(<span class="keyword">auto</span>&amp;&amp; s: selected)</div>
<div class="line"><a id="l00298" name="l00298"></a><span class="lineno">  298</span>            s = <span class="keyword">false</span>;</div>
<div class="line"><a id="l00299" name="l00299"></a><span class="lineno">  299</span>        </div>
<div class="line"><a id="l00300" name="l00300"></a><span class="lineno">  300</span>        <span class="comment">//create front using &quot;fake ref&quot;</span></div>
<div class="line"><a id="l00301" name="l00301"></a><span class="lineno">  301</span>        std::vector&lt;Point&gt; front = createFront(points, 0,0);</div>
<div class="line"><a id="l00302" name="l00302"></a><span class="lineno">  302</span>            </div>
<div class="line"><a id="l00303" name="l00303"></a><span class="lineno">  303</span>        <span class="comment">//get reference value from extremal points</span></div>
<div class="line"><a id="l00304" name="l00304"></a><span class="lineno">  304</span>        <span class="keywordtype">double</span> refX= front.back().f1;</div>
<div class="line"><a id="l00305" name="l00305"></a><span class="lineno">  305</span>        <span class="keywordtype">double</span> refY= front.front().f2;</div>
<div class="line"><a id="l00306" name="l00306"></a><span class="lineno">  306</span>            </div>
<div class="line"><a id="l00307" name="l00307"></a><span class="lineno">  307</span>        <span class="keywordflow">for</span>(<span class="keyword">auto</span>&amp;&amp; point: front){</div>
<div class="line"><a id="l00308" name="l00308"></a><span class="lineno">  308</span>            point.f1 -= refX;</div>
<div class="line"><a id="l00309" name="l00309"></a><span class="lineno">  309</span>            point.f2 -= refY;</div>
<div class="line"><a id="l00310" name="l00310"></a><span class="lineno">  310</span>        }</div>
<div class="line"><a id="l00311" name="l00311"></a><span class="lineno">  311</span>        </div>
<div class="line"><a id="l00312" name="l00312"></a><span class="lineno">  312</span>        <span class="comment">//mark the extrema as chosen and remove them from the front</span></div>
<div class="line"><a id="l00313" name="l00313"></a><span class="lineno">  313</span>        selected[front.front().index] = <span class="keyword">true</span>;</div>
<div class="line"><a id="l00314" name="l00314"></a><span class="lineno">  314</span>        selected[front.back().index] = <span class="keyword">true</span>;</div>
<div class="line"><a id="l00315" name="l00315"></a><span class="lineno">  315</span>        front.pop_back();</div>
<div class="line"><a id="l00316" name="l00316"></a><span class="lineno">  316</span>        front.erase(front.begin(),front.begin()+1);</div>
<div class="line"><a id="l00317" name="l00317"></a><span class="lineno">  317</span>        <span class="keywordflow">if</span>(k == 2) <span class="keywordflow">return</span>;</div>
<div class="line"><a id="l00318" name="l00318"></a><span class="lineno">  318</span>        </div>
<div class="line"><a id="l00319" name="l00319"></a><span class="lineno">  319</span>        <span class="comment">//find the optimal set in the front. afterwards selected points have selected=true</span></div>
<div class="line"><a id="l00320" name="l00320"></a><span class="lineno">  320</span>        hypSSP(front,k-2);</div>
<div class="line"><a id="l00321" name="l00321"></a><span class="lineno">  321</span>        <span class="comment">//mark selected points in the original front</span></div>
<div class="line"><a id="l00322" name="l00322"></a><span class="lineno">  322</span>        <span class="keywordflow">for</span>(Point <span class="keyword">const</span>&amp; point: front){</div>
<div class="line"><a id="l00323" name="l00323"></a><span class="lineno">  323</span>            <span class="keywordflow">if</span>(point.selected){</div>
<div class="line"><a id="l00324" name="l00324"></a><span class="lineno">  324</span>                selected[point.index] = <span class="keyword">true</span>;</div>
<div class="line"><a id="l00325" name="l00325"></a><span class="lineno">  325</span>            }</div>
<div class="line"><a id="l00326" name="l00326"></a><span class="lineno">  326</span>        }</div>
<div class="line"><a id="l00327" name="l00327"></a><span class="lineno">  327</span>    }</div>
</div>
<div class="line"><a id="l00328" name="l00328"></a><span class="lineno">  328</span>};</div>
</div>
<div class="line"><a id="l00329" name="l00329"></a><span class="lineno">  329</span> </div>
<div class="line"><a id="l00330" name="l00330"></a><span class="lineno">  330</span>}</div>
<div class="line"><a id="l00331" name="l00331"></a><span class="lineno">  331</span><span class="preprocessor">#endif</span></div>
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